What does Acts 21:25 mean?
Paul is speaking with James and the elders of the church in Jerusalem. He intends to update them on what he has been doing the last four years or so. His work included planting a church in Ephesus and building up mostly-Gentile churches all over the coast of the Aegean Sea. But a rumor has been floating around Jerusalem that Paul is teaching Jewish Christians they should not observe the Mosaic law. This is false, and Paul is willing to take the elders' suggestion—help four men complete a Jewish vow—to debunk the unhelpful gossip (Acts 21:18–24).It's not clear why the elders tell him this. These regulations came about because the apostles and elders had to consider if Gentile believers had to become practicing Jews to follow Jesus. Paul was there. In fact, Paul brought them the issue from Syrian Antioch. And Paul was responsible to take their letter to the Gentiles in Syria and modern-day Turkey (Acts 15). Perhaps they're reassuring him that their initial request to the Gentiles stands.
"What has been sacrificed to idols" refers to the communal meals people have in pagan temples. To refrain means more than just not going out to dinner; during those meals people make business connections and show their fealty to their city. A person who doesn't sacrifice to the local god or goddess is seen as someone who doesn't care about their community. Without business contacts, a craftsman can suffer financially and find it difficult to provide for his family. It's a big sacrifice, but Paul supports it; false gods are demons, and Jesus-followers should have no part (1 Corinthians 10:14–22).
The council forbids animals that have been strangled because such animals are not butchered correctly to ensure their blood is properly drained. There is debate today about whether Christians are still prohibited from eating blood because the law pre-dates the Mosaic law (Genesis 9:4) or if the restriction was lifted the same time other foods were (Acts 10:9–16). Either way, the council asks the Gentiles to refrain so the Jewish Christians will feel free to share meals.
"Sexual immorality" means any sex not between a husband and wife. There is no caveat for couples who are engaged, couples who "love each other very much," or pornographic situations. Such restrictions have always been in place, and remain; that they need to be repeated so often speaks to the powerful temptation of sex.